1. Field of the Invention
The invention is directed to reflective surfaces capable of providing non-reversed, substantially undistorted reflections and a method for forming said reflective surfaces. The invention may be used as a novelty item, toy, or a mirror to facilitate various activities, such as shaving. Alternatively, the invention may have a wide variety of applications in the field of optics.
2. Description of the Related Technology
It is known that by curving or creating a reflective surface with a non-linear contour, it is possible to enlarge or alter a reflection. The reflected images of these non-planar mirrors, however, are generally significantly distorted, producing non-perspective projections or poor perspective projections depending upon the curvature and shape of the mirror. By contrast, a substantially undistorted reflected image, producing a perspective projection is formed by tracing a line from an image plane I through a point COP, known as the center of projection or focal point, until it touches an object in the scene or object plane S, as shown in FIG. 1. This method of image formation can be physically realized by using a pinhole camera, where the pinhole plays the role of the center of projection. In a pinhole camera the film, or image plane I, lies behind the pinhole, but the resulting image is geometrically similar to one formed by placing a virtual image plane in front of the pinhole, as shown in FIG. 1.
Curved rectifying mirrors of the prior art that utilize this pinhole camera concept, such as the mirrors described in Hicks et al., “Reflective Surfaces as Computational Sensors,” Image and Vision Computing, Volume 19, Issue 11, September 2001, pages 773-777, are capable of minimizing image distortion This mirror, however, produces a wide angle, conventional reversed image.
Hicks, et al., “Geometric distributions and catadioptric sensor design,” IEEE Computer Society Conference on Computer Vision Pattern Recognition (2001), discloses a non-reversing mirror capable of producing a non-reversed and approximately rectified image of an object only when the reflective surface is tilted 45° with respect to the optical axis of the observer. Although the mirror produces a non-reversing and relatively undistorted side view reflection, it is not capable of projecting a substantially undistorted direct reflection when an observer is positioned within the field of view of the mirror.
Another mirror described in Hicks et al., “The method of vector fields for catadioptric sensor design with applications to panoramic imaging”, IEEE Computer Society Conference on Computer Vision Pattern Recognition, (2004), projects a non-reversed minimally distorted reflection when the viewer is positioned at an infinite or extremely large distance from the mirror. However, the mirror is not designed to project a substantially undistorted reflection when the viewer is positioned relatively close to the mirror.
Thomas describes in “Mirror Images”, Scientific American, December 1980, pp. 206-22, a non-reversing mirror made from a portion of a torus of revolution, i.e. a “toroidal” surface, more commonly referred to as a “donut” shape. The equation for a torus of revolution, having major radius a, and minor radius r, is:
                                                        a              4                        2                    +                                    r              4                        2                    +                                    a              2                        ⁢                          z              2                                -                                    a              2                        ⁢                          r              2                                -                                    y              2                        ⁢                          r              2                                -                                    y              2                        ⁢                          a              2                                -                                    x              2                        ⁢                          y              2                                +                                    x              4                        2                    +                                    y              2                        ⁢                          x              2                                +                                    z              2                        ⁢                          x              2                                -                                    x              2                        ⁢                          a              2                                +                                    y              2                        ⁢                          z              2                                +                                    y              4                        2                    +                                    z              4                        2                    -                                    z              2                        ⁢                          r              2                                      =        0                            Equation        ⁢                                  ⁢        1            wherein all points (x,y,z) in three dimensional space that satisfy Equation 1 will form a surface of a torus of revolution, as shown in FIG. 2, and wherein a and r constrains the mirror size and the distance from the observer to the mirror.
This mirror only projects an undistorted image when a fixed reflective plane occupying a certain position in space and reflecting an object plane positioned at a certain position in space is viewed from a pre-determined distance.
The mathematical expression for this mirror contains at most only fourth degree terms, which means that there can be substantial image distortion. In general, optical rectification requires higher order terms to produce an undistorted image. To design a non-reversing mirror having unit magnification, higher order terms are necessary to minimize distortion and/or to allow the designer to choose the optimal distance at which the mirror will be used. Thus, equation 1 does not permit a designer to create a mirror capable of projecting a non-reversing and non-distorting reflection for a pre-selected distance between the mirror and object plane to be reflected.
Another deficiency of the Thomas mirror is its inability to incorporate scaling constants in its design. Consequently, it is not possible to scale a reflection in the vertical and horizontal directions or project a substantially undistorted reflection when the mirror is viewed from different angles. The mirror, therefore, provides no means for controlling, minimizing or eliminating distortions in the vertical and horizontal directions and has been found to generate undesirable and substantial image distortions at unit magnification and at lower magnifications. Some image distortions are also produced at magnifications higher than unit magnification.
In U.S. Pat. No. 4,116,540, Thomas describes another non-reversing mirror known as a “monkey saddle.” This mirror, however, does not allow for the incorporation of scaling constants and thus, produces substantial image distortion. The distortion is evident at all magnifications, but is particular prominent at unit magnification or lower magnifications. In view of the aforementioned design deficiencies, there is a need to develop a reflective surface capable of projecting a non-reversing and substantially undistorted direct reflection at various magnifications.